To a variety over a characteristic zero field, one can associate its compactly supported $\mathbb{A}^1$-Euler characteristic using motivic homotopy theory, as was done in work of Arcila-Maya, Bethea, Opie, Wickelgren and Zakharevich. These are quadratic forms, which carry a lot of information, but they are often difficult to compute in practice. I will discuss joint work in progress with Jesse Pajwani and Herman Rohrbach in which we apply the machinery of power structures on the Grothendieck-Witt ring as introduced in work of Pajwani and Pál to calculate the compactly supported $\mathbb{A}^1$-Euler characteristic of the symmetric powers of a cellular variety.